To solve the equation [tex]\(\ln 2 + \ln (x + 2) = -2\)[/tex] for [tex]\(x\)[/tex], follow these step-by-step instructions:
1. Combine the logarithmic terms on the left side of the equation using the property of logarithms that states [tex]\(\ln a + \ln b = \ln(ab)\)[/tex]:
[tex]\[
\ln(2) + \ln(x + 2) = \ln(2(x + 2))
\][/tex]
2. Substitute this back into the equation, resulting in:
[tex]\[
\ln(2(x + 2)) = -2
\][/tex]
3. Exponentiate both sides to eliminate the natural logarithm (remembering that [tex]\(e^{\ln y} = y\)[/tex]):
[tex]\[
2(x + 2) = e^{-2}
\][/tex]
4. Solve for [tex]\(x\)[/tex] by isolating [tex]\(x\)[/tex]. First, divide both sides by 2:
[tex]\[
x + 2 = \frac{e^{-2}}{2}
\][/tex]
5. Subtract 2 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{e^{-2}}{2} - 2
\][/tex]
6. Simplify the right side of the equation:
[tex]\[
x = -2 + \frac{e^{-2}}{2}
\][/tex]
Thus, the solution to the equation [tex]\(\ln 2 + \ln (x + 2) = -2\)[/tex] is:
[tex]\[
x = -2 + \frac{e^{-2}}{2}
\][/tex]
